Numerical solution of a nonsteady differential equation of heat conduction

The use of floating weight is suggested in the numerical solution of a parabolic differential equation of heat conduction with variable coefficients in integral-mean temperatures, used in the calculation of thermal expansions of turbine components. Recommendations are given for the determination of the optimum weights.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 2, pp. 319–327, February, 1978.     

Numerical solution of a parabolic equation

A splitting scheme is proposed for the numerical solution of a parabolic equation containing singularities important on the boundary of the region of the sought function.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 6, pp. 1090–1092, June, 1981.     

Numerical solution of a nonlinear Poisson equation

A method that is convenient for practical applications is proposed for solving the nonlinear Poisson equation.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 36, No. 6, pp. 1077–1079, June, 1979.     

Numerical solution to a sideways parabolic equation

The inverse heat conduction problem can be considered to be a sideways parabolic equation in the quarter plane. This is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but one side is inaccessible to measurements. A numerical procedure for this severely ill‐posed problem is suggested, which consists of two steps, namely a mollification of the data and a marching difference scheme. The numerical method is proved to be stable. Several computational results are presented and discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

On a bounded solution to a pair of coupled nonlinear ordinary differential equations

We present a bounded, decaying solution to a pair of coupled, nonlinear second-order ordinary differential equations arising in the theory of natural convection. The solution is found by transforming the problem into a non-autonomous system in the phase-plane. A uniqueness proof is given for the bounded solution.     

On a bounded solution to a pair of coupled nonlinear ordinary differential equations

We present a bounded, decaying solution to a pair of coupled, nonlinear second-order ordinary differential equations arising in the theory of natural convection. The solution is found by transforming the problem into a non-autonomous system in the phase-plane. A uniqueness proof is given for the bounded solution.     

Numerical solution of Lippmann-Schwinger integral equation

A method is discussed for obtaining a numerical solution of an equation similar to the equation for the transport of radiant energy for a steady radiation field [1].Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 2, pp. 329–331, August, 1977.     

Efficient algorithms for ordinary differential equation model identification of biological systems

Algorithms for parameter estimation and model selection that identify both the structure and the parameters of an ordinary differential equation model from experimental data are presented. The work presented here focuses on the case of an unknown structure and some time course information available for every variable to be analysed, and this is exploited to make the algorithms as efficient as possible. The algorithms are designed to handle problems of realistic size, where reactions can be nonlinear in the parameters and where data can be sparse and noisy. To achieve computational efficiency, parameters are mostly estimated for one equation at a time, giving a fast and accurate parameter estimation algorithm compared with other algorithms in the literature. The model selection is done with an efficient heuristic search algorithm, where the structure is built incrementally. Two test systems are used that have previously been used to evaluate identification algorithms, a metabolic pathway and a genetic network. Both test systems were successfully identified by using a reasonable amount of simulated data. Besides, measurement noise of realistic levels can be handled. In comparison to other methods that were used for these test systems, the main strengths of the presented algorithms are that a fully specified model, and not only a structure, is identified, and that they are considerably faster compared with other identification algorithms.     

An ordinary differential equation for the Green function of time-domain free-surface hydrodynamics

In this paper a general fourth-order ordinary differential equation is derived for a class of functions including the time-domain Green function of linearized free-surface hydrodynamics and all its spatial derivatives. Among all the applications following from this new result, the acceleration of numerical computations in BEM solutions of time-domain hydrodynamics was the initial motivation of this work. Two new alternative methods for the computation of convolution integrals based on the new ODEs are suggested and illustrated by a numerical example.     

A finite element method for the general solution of ordinary differential equations

A finite element method is developed by which it is possible to obtain the general solution of an ordinary differential equation directly. The procedure consists of approximating the differential equation with a rectangular matrix equation and of solving the latter equation by using generalized matrix inversion. It is shown in the paper that the homogeneous and inhomogeneous solutions of the two systems correspond and that the approximate solutions produced form the complete general solution of the original differential equation.     

Numerical solution of dispersion equation in one dimension

Abstract

An improved hybrid method for one‐dimensional advection‐diffusion problems, based on the Holly‐Preissmann two‐point fourth‐order and Crank‐Nicholson numerical schemes, has been proposed to handle the problem with Courant numbers (Cr) greater than 1. Extensive test runs and analyses have been performed for a schematic advection‐diffusion problem. Through a comparison of the analytical solution with the computed results, the accuracy and stability of this improved hybrid method are discussed. Satisfactory results are found for both weak and strong diffusion problems under large Courant number conditions. The sensitivity of the improved method to the temporal weighting factor has also been demonstrated. For strong diffusion problems, the use of a larger temporal weighting factor becomes necessary to eliminate the phenomenon of instability.     

A note on the solution of a differential equation arising in boundary-layer theory

Summary The differential equation f + ff + f² = 0 (where dashes denote differentiation with respect to the independent variable ) subject to the boundary conditions f(0)=0, f()=0 and either f(0)=1 or f(0)=–1 is considered. It is shown that by using pf as dependent variable and =C–f (where C=f()) as independent variable and then expanding in powers of , a very good approximation to the solution can be obtained using only a few terms in the expansion.     

Numerical solution of the Helmholtz equation with high wavenumbers

This paper investigates the pollution effect, and explores the feasibility of a local spectral method, the discrete singular convolution (DSC) algorithm for solving the Helmholtz equation with high wavenumbers. Fourier analysis is employed to study the dispersive error of the DSC algorithm. Our analysis of dispersive errors indicates that the DSC algorithm yields a dispersion vanishing scheme. The dispersion analysis is further confirmed by the numerical results. For one‐ and higher‐dimensional Helmholtz equations, the DSC algorithm is shown to be an essentially pollution‐free scheme. Furthermore, for large‐scale computation, the grid density of the DSC algorithm can be close to the optimal two grid points per wavelength. The present study reveals that the DSC algorithm is accurate and efficient for solving the Helmholtz equation with high wavenumbers. Copyright © 2003 John Wiley & Sons, Ltd.     

Decomposition of ordinary differential polynomials

In this paper, we present a complete algorithm to decompose nonlinear differential polynomials in one variable and with coefficients in a computable differential field of characteristic zero. The algorithm provides an efficient reduction of the problem to the factorization of LODOs over the same coefficient field. Besides arithmetic operations, the algorithm needs decomposition of algebraic polynomials, factorization of multi-variable polynomials, and solution of algebraic linear equation systems. The algorithm is implemented in Maple for the constant field case. The program can be used to decompose differential polynomials with thousands of terms effectively. This article was partially supported by a National Key Basic Research Project of China (NO. G1998030600) and by a USA NSF grant CCR-0201253.     

Finite-difference solution of an elliptic partial differential equation with discontinuous coefficients

An algoritm is presented for point relaxation of an elliptic partial diferential equation at a grid station where its coefficient are discontinuous. Such equations can arise from the use of skewed co-ordinate system based on discotinuous reference functions (lengths) to map a complicated physical geometry onto a simple computational domain. Efficary of the scheme is illustrated by examples involving potential flow in a sharply bent channel and in a conical diffuser.     

Numerical solution of the complete mass balance equation in chromatography

A new approach to simulate the movement of bands through a chromatographic column is presented. Similar to the Craig distribution model, the mass balance equation is divided into two equations describing two successive processes. The first equation includes two effects: solute diffusion in the mobile phase and migration of the solute band with the mobile phase. The second equation deals with the distribution of the solute between phases, i.e., the adsorption isotherm. The partial differential equations are integrated numerically over time and space using two methods. The first approach is a finite difference method. In the second approach, the propagation operator is expanded in a Chebyshev series, where large time steps can be used. The rate of adsorption and desorption is determined by the size of the time increment. By varying the size of the time step, it is possible to study kinetic effects. The influences of sample size, injection width, rate of mass transfer, and mobile phase velocity on the elution profile were studied. Simulations using the modified Craig approach with either of the two numerical procedures showed that the solutes behaved in the chromatographically expected manner. Moreover, with linear adsorption isotherms, direct relationships between HETP, as well as retention times, and experimental parameters could be established.     

Chebyshev matrix operator method for the solution of integrated forms of linear ordinary differential equations

Summary This article reports a method to handle integrated forms of linear ordinary differential equations by means of matrix operator expressions, which apply to integral terms and non-constant coefficients. The method avoids the use of a numerical grid and includes the treatment of boundary conditions and inhomogeneous terms. It can be regarded as a mechanized version of the -method. The application of the method to equations in integrated form leads to linear algebraic systems with better condition compared to the differential operator. Therefore, the method permits the application of iterative methods in order to solve the linear systems. Its effective application is demonstrated by two examples. Furthermore, the method is extended to linear parabolic problems. Finally, a solution of the Orr-Sommerfeld equation is presented to indicate the treatment of eigenvalue problems.     

Numerical solution of the variation boundary integral equation for inverse problems

In this paper a procedure to solve the identification inverse problems for two‐dimensional potential fields is presented. The procedure relies on a boundary integral equation (BIE) for the variations of the potential, flux, and geometry. This equation is a linearization of the regular BIE for small changes in the geometry. The aim in the identification inverse problems is to find an unknown part of the boundary of the domain, usually an internal flaw, using experimental measurements as additional information. In this paper this problem is solved without resorting to a minimization of a functional, but by an iterative algorithm which alternately solves the regular BIE and the variation BIE. The variation of the geometry of the flaw is modelled by a virtual strainfield, which allows for greater flexibility in the shape of the assumed flaw. Several numerical examples demonstrate the effectiveness and reliability of the proposed approach. Copyright © 2000 John Wiley & Sons, Ltd.